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Density dependent replicator-mutator models in directed evolution

  • * Corresponding author: Matthieu Alfaro

    * Corresponding author: Matthieu Alfaro 
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  • We analyze a replicator-mutator model arising in the context of directed evolution [24], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [14] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena.

    Mathematics Subject Classification: Primary: 35B40, 35B09; Secondary: 92B05.

    Citation:

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  • Figure 1.  Evolution of Gaussian solutions for $ {\sigma ^2} = 1 $, $ a_0 = 1 $ and (from left to right) $ m_0 = -4 $, $ m_0 = 0 $ and $ m_0 = 4 $

    Figure 2.  (A): The first four approximations, for $ 0\leq t\leq 3 $, of the nonlocal term $ \overline{u}(t) $ computed via the fixed point iteration (28). (B): Numerical solution obtained by the method described in Section 5, starting from $ u_0 = \mathbb{1}_{[1/2,3/2]} $, with $ {\sigma ^2} = 1 $. The red points are the points on the graph $ u(t,\cdot) $ with abscissa $ x = \overline u(t) $. The green points are the maxima of $ u(t,\cdot) $. This reveals the dissymmetry of the solution

    Figure 3.  Vector field defined by the differential system (31) with $ {\sigma ^2} = 1 $, describing the dynamics of Gaussian solutions. In yellow, the set of initial conditions for which $ a $ blows up in finite time $ T^{\star} $ and in red, dark blue and light blue those for which both $ a $ and $ m $ are globally defined. The red dashed curve is the set of values defined by $ m_0 = -1/{(a_0\sqrt{2{\sigma ^2}})} $, for which $ a $ tends to infinity and $ m $ tends to zero as time goes to infinity. The dark blue region corresponds to the values leading to an anti-diffusion/diffusion behaviour. The light blue region corresponds to the values leading to a pure diffusion behaviour

    Figure 4.  Case $ (i) $ concentration in finite time. The values of the parameters are $ a_0 = 5/64 $, $ m_0 = -585/64 $, $ {\sigma ^2} = 1 $. It follows that $ T^\star\approx 1.878 $ and $ m(T^\star)\approx -1.277<0 $

    Figure 5.  Case $ (ii) $ concentration in infinite time: the solution converges to a Dirac mass at zero. The values of the parameters are $ a_0 = 3/16 $, $ m_0 = -8\sqrt{2}/3 $, $ {\sigma ^2} = 1 $

    Figure 6.  Case $ (iii) $ with $ m_0<0 $, anti-diffusion/diffusion behaviour. The values of the parameters are $ a_0 = 2/10 $, $ m_0 = -34/10 $, $ {\sigma ^2} = 1 $

    Figure 7.  Case $ (iii) $ with $ m_0\geq 0 $, the solution is flattening and accelerating. The values of the parameters are $ a_0 = 3/2 $, $ m_0 = 7/2 $, $ {\sigma ^2} = 1 $

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